Torques acting on a cylinder, with friction

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The discussion focuses on a problem involving a rolling cylinder that appears to be dragging a rod with a plate experiencing friction. Participants express the need for a detailed description of the problem rather than assumptions. There is confusion regarding the meaning of labeled arrows in the diagram, particularly the curvy arrows labeled ##\beta## and ##\omega##. A torque equation is mentioned, but its derivation is unclear without additional context. Clarity on these points is essential for a proper understanding of the mechanics involved.
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Homework Statement
I need to understand where the very first equation comes from?
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torque?
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Please describe the problem in detail not just part of the proffered solution. What is going on here? It looks like you have a rolling cylinder that is dragging a rod at the end of which is a plate with friction. However, this is only a guess and we do not like guessing if it can be avoided.

Furthermore, what do all these labeled arrows represent, especially the curvy arrows labeled ##\beta## and ##\omega##? Again, we do not like guessing.

That said, yes, the first equation looks like a torque equation. Where it comes from depends on details that we do not have.
 
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Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .

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